The value of k for which straight line x + y + 3z - 2 = 0 = 2x + y - z - 3 is parallel to the plane 3x + 2y + kz - 4 = 0 is:

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Question

This question was previously asked in

AAI ATC Junior Executive 21 Feb 2023 Shift 2 Official Paper

Answer 1

Given:

The straight line x + y + 3z - 2 = 0 = 2x + y - z - 3 is parallel to the plane 3x + 2y + kz - 4 = 0

Concept:

If a line parallel to the plane then dot product of direction ratio of line and normal to plane must be 0

Calculation:

The straight line x + y + 3z - 2 = 0 = 2x + y - z - 3 is parallel to the plane 3x + 2y + kz - 4 = 0

So, the direction ratios of normals to the planes are ( 1, 1, 3 ) and ( 2, 1, -1 ) repectively .

So direction ratio of the required line is the cross product of these normals

i.e. \(\rm \begin{vmatrix}i&j&k\\1&1&3\\2&1&-1\end{vmatrix}\)

= i(-1 - 3) - j(-1 - 6) + k(1 - 2)

= - 4i + 7j - k

Therefore, direction ratio of line is ( -4, 7, -1)

Now line is parallel to 3x + 2y + kz - 4 = 0, So dot product of direction ratio of line and normal to plane must be 0

∴ 3(-4) + 2(7) + k(-1) = 0

⇒ 2 - k = 0

⇒ k = 2

Hence the option (2) is correct.

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